Welcome to trigonometry! Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. It's most commonly introduced using right-angled triangles, which have one angle of 90 degrees. In a right triangle, we name the sides relative to an angle: the hypotenuse is the longest side opposite the right angle, the opposite side is directly across from our angle of interest, and the adjacent side is next to our angle but not the hypotenuse.
The foundation of trigonometry is built on three main ratios: sine, cosine, and tangent. These ratios relate the sides of a right triangle to its angles. Sine, abbreviated as sin, equals the opposite side divided by the hypotenuse. Cosine, or cos, equals the adjacent side divided by the hypotenuse. Tangent, or tan, equals the opposite side divided by the adjacent side. A helpful mnemonic to remember these is SOH-CAH-TOA: Sine equals Opposite over Hypotenuse, Cosine equals Adjacent over Hypotenuse, and Tangent equals Opposite over Adjacent.
Another powerful way to understand trigonometric functions is through the unit circle. The unit circle is a circle with radius 1 centered at the origin of a coordinate system. For any angle theta, we can draw a radius from the origin to a point on the circle. The x-coordinate of this point equals cosine theta, and the y-coordinate equals sine theta. The tangent of theta equals the y-coordinate divided by the x-coordinate, which is the same as sine theta divided by cosine theta. As the angle increases, the point moves counterclockwise around the circle, and the sine and cosine values change accordingly.
Trigonometric functions can be visualized as graphs, showing their values as the angle changes. The sine function oscillates smoothly between -1 and 1, starting at 0, reaching 1 at π/2, returning to 0 at π, dropping to -1 at 3π/2, and back to 0 at 2π. The cosine function follows a similar pattern but shifted by π/2 radians, starting at 1 when the angle is 0. Both functions repeat every 2π radians, which is 360 degrees, making them periodic functions. The tangent function, not shown here, has vertical asymptotes at π/2, 3π/2, and so on, where cosine equals zero.
To summarize what we've learned about trigonometry: First, trigonometry is the study of relationships between angles and sides in triangles, particularly right triangles. The three fundamental trigonometric ratios are sine, cosine, and tangent, which we remember using SOH-CAH-TOA. The unit circle gives us another way to understand these functions, where sine is the y-coordinate and cosine is the x-coordinate of a point on the circle. Trigonometric functions are periodic, repeating every 2π radians or 360 degrees. These concepts have countless applications in fields like navigation, engineering, physics, architecture, and computer graphics.